The present invention relates to a golf ball, and more particularly, to the cross-sectional profile of dimples on the surface of a golf ball.
Golf balls were originally made with smooth outer surfaces. In the late nineteenth century, players observed that the guttie golf balls traveled further as they got older and more gouged up. The players then began to roughen the surface of new golf balls with a hammer to increase flight distance. Manufacturers soon caught on and began molding non-smooth outer surfaces on golf balls.
By the mid 1900""s , almost every golf ball being made had 336 dimples arranged in an octahedral pattern. Generally, these balls had about 60 percent of their outer surface covered by dimples. Over time, improvements in ball performance were developed by utilizing different dimple patterns. In 1983, for instance, Titleist introduced the TITLEIST 384, which, not surprisingly, had 384 dimples that were arranged in an icosahedral pattern. About 76 percent of its outer surface was covered with dimples. Today""s dimpled golf balls travel nearly two times farther than a similar ball without dimples.
The dimples on a golf ball are important in reducing drag and increasing lift. Drag is the air resistance that acts on the golf ball in the opposite direction from the ball flight direction. As the ball travels through the air, the air surrounding the ball has different velocities and, thus, different pressures. The air exerts maximum pressure at the stagnation point on the front of the ball. The air then flows over the sides of the ball and has increased velocity and reduced pressure. At some point it separates from the surface of the ball, leaving a large turbulent flow area called the wake that has low pressure. The difference in the high pressure in front of the ball and the low pressure behind the ball slows the ball down. This is the primary source of drag for a golf ball.
The dimples on the ball create a turbulent boundary layer around the ball, i.e., the air in a thin layer adjacent to the ball flows in a turbulent manner. The turbulence energizes the boundary layer and helps it stay attached further around the ball to reduce the area of the wake. This greatly increases the pressure behind the ball and substantially reduces the drag.
Lift is the upward force on the ball that is created from a difference in pressure on the top of the ball to the bottom of the ball. The difference in pressure is created by a warpage in the air flow resulting from the ball""s back spin. Due to the back spin, the top of the ball moves with the air flow, which delays the separation to a point further aft. Conversely, the bottom of the ball moves against the air flow, moving the separation point forward. This asymmetrical separation creates an arch in the flow pattern, requiring the air over the top of the ball to move faster, and thus have lower pressure than the air underneath the ball.
Almost every golf ball manufacturer researches dimple patterns in order to increase the distance traveled by a golf ball. A high degree of dimple coverage is beneficial to flight distance, but only if the dimples are of a reasonable size. Dimple coverage gained by filling spaces with tiny dimples is not very effective, since tiny dimples are not good turbulence generators.
In addition to researching dimple pattern and size, golf ball manufacturers also study the effect of dimple shape, volume, and cross-section on overall flight performance of the ball. One example is U.S. Pat. No. 5,737,757, which discusses making dimples using two different spherical radii with an inflection point where the two curves meet. In most cases, however, the cross-sectional profiles of dimples in prior art golf balls are parabolic curves, ellipses, semi-spherical curves, saucer-shaped, a sine curve, a truncated cone, or a flattened trapezoid. One disadvantage of these shapes is that they can sharply intrude into the surface of the ball, which may cause the drag to become greater than the lift. As a result, the ball may not make best use of momentum initially imparted thereto, resulting in an insufficient carry of the ball. Despite all the cross-sectional profiles disclosed in the prior art, there has been no disclosure of a golf ball having dimples defined by the revolution of a catenary curve.
The present invention is directed to defining dimples on a golf ball by revolving a catenary curve about its symmetrical axis. In one embodiment, the catenary curve is defined by a hyperbolic sine function. In another embodiment, the catenary curve is defined by a hyperbolic cosine function. In a preferred embodiment, the catenary curve used to define a golf ball dimple is a hyperbolic cosine function in the form of:   Y  =            d      ⁢              (                              cosh            ⁢                          (                              a                ⁢                                  xe2x80x83                                ⁢                x                            )                                -          1                )                            cosh        ⁢                  (                      a            ⁢                          xe2x80x83                        ⁢            r                    )                    -      1      
where:
Y is the vertical distance from the dimple apex,
x is the radial distance from the dimple apex,
a is the shape constant;
d is the depth of the dimple, and
r is the radius of the dimple.